Using the weak positivity restriction placed on the boundary, together with the logarithmic balancing condition of stable log maps, we first show that certain types of subtrees of rational curves cannot appear in curves \(f: C \rightarrow X\)  associated to a point of \(\mathcal {M}(X,\beta )\) . Specifically, letting \(\tau = (G_\tau ,\sigma ,\textbf {u})\)  be the tropical type of \(C\) , we claim that if \(v \in V(G_\tau )\)  is connected to a vertex \(v_0\)  by an edge \(e\)  such that \(\sigma (v_0) = 0 \in \Sigma (X)\) , and \(\sigma (v)\ne  0\) , then \(v\)  is contained in the spine of \(G_\tau \) . First, let \(D^{\prime } = \sum _{i} a_iD_i\) , \(a_i >0\) , be the nef divisor supported on \(D\) , and suppose \(v\)  was not contained in the spine of \(G_\tau \) . Letting \(q_e \in C\)  be the node associated to the edge \(e\) , consider the closure of the component of the \(C \setminus \lbrace q_e\rbrace \)  which contains contains the generic point of the \(C_v \subset C\) . By restricting the log structure on \(C\)  to this union of components, we produce a punctured log curve \(C^{\prime } \rightarrow X\) . Since \(v\)  was not contained in the spine of \(G_\tau \) , \(C^{\prime }\)  has only one punctured point corresponding to the former node \(q_e\) . Since \(\sigma (v_0) = 0\)  but \(\sigma (v) \ne  0\) , we must have \(\textbf {u}(e)(D_i)\le 0\)  for all \(i\) , and \(\textbf {u}(D_j)<0\)  for some \(j\) . Thus, by applying the logarithmic balancing condition of [1]} Proposition \(2.27\) , we find:
\(C^{\prime } \cdot D^{\prime } = \sum _i a_i \textbf {u}(e)(D^{\prime }) < 0.\) 
